Teaching materials · Cycle 2
Combinatorial Games & Invariants.
Three live sessions on invariants, combinatorial game theory, and cellular automata, closed in January 2026 with teacher presentations by cohort teachers on Towers of Hanoi, Nim, a Japanese puzzle, and an introduction to game theory.
Coach: Ilja Uzelac Bujišić. Three live sessions took place on 28 November, 12 December, and 26 December 2025. The cycle closed on 16 January 2026 with teacher presentations.
What the cycle covered
- 28 November 2025. Invariants, monovariants, parity and colorings.
- 12 December 2025. Combinatorial game theory, winning and losing positions, Nim.
- 26 December 2025. A special extension connected Nim variations with cellular automata, Conway's Game of Life, simulation, and mathematical research.
- 16 January 2026, cohort teacher presentation. Teachers presented on the Towers of Hanoi, the game of piles (Nim), a Japanese puzzle, and an introduction to game theory.
The cycle was built around the idea that a good invariant, something that doesn't change under a move, often does more work than a full case analysis, and that the same instinct applies to reachability arguments in games. The teacher presentation put that idea in cohort teachers' own words.
The 26 December description draws on the Zoom AI Companion session summary supplied by the Academy. It presents a secondary synopsis of the session.
Exercise collection
The problems from Cycle II.
The full exercise collection worked across this cycle, kept as a single document so the mathematics can be revisited in its original form.
These are the exercises used in the cycle's sessions. Individual problems are drawn from the olympiad problem literature and compiled for the Academy's teaching. External publications are listed on the Resources page.